rev101b - 1. Diversification A stock market speculator is...

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1 1 1. Diversification A stock market speculator is considering two investment opportunities x An investment in the first company would yield an income of 50,000 in a strong economy and an income of 10,000 in a recession. x The second company is much more recession proof and yield 30,000 in a strong economy, but 20,000 in a recession. x The investor believes that the chances of a strong economy are 50/50. x The investor's utility function is U(Y)=ln Y, where Y is the investor's income from his investment. 2 1. Diversification continued a) Suppose that he must choose one company or the other. Which investment has the greater expected yield? Which investment has the greater expected utility for the speculator? x For firm 1, EU=.5 ln 50000 +.5 ln 10000 = 10.015, EV=.5*50000+.5*10000=30000. x For firm 2, EU=.5 ln 30000 +. 5ln 20000 = 10.106, EV=.5*30000+.5*20000=25000 x Firm 1 is better in expected value terms, but firm 2 is better in expected utility terms, since the individual is risk averse.
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2 3 1. Diversification continued b) Without a forecaster, suppose that the speculator can diversify and spread his funds equally between the two firms. Will he choose to invest in both firms? EU=.5 ln 40000 +.5 ln 15000 = 10.106, so he is pretty much indifferent between a 50/50 split and buying only firm 2. 4 1. Diversification continued c) What mix of firm 1 and firm 2 would maximize expected utility to the speculator? x EU=.5 ln ( α 50000 + (1- α )30000) +.5 ln ( α 10000 + (1- α )20000) x EU/ ∂α =10000/(20000 α +30000) - 5000/(-10000 α +20000) x α =.25, so he will invest 25 percent in firm1 and 75 percent in firm 2
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3 5 2. Risk Aversion x Luke is a miner living in a poor third world country. After 20 years of prospecting, he gets lucky and discovers $3 million in gold (his only wealth in the world). He wants to get the gold to the capital city, so he can retire and live off his wealth. x Over the years, Luke has discovered that local bandits steal his money on every other trip to the capital. x Luke considers two strategies: Strategy A: He makes one trip and takes all the money. Strategy B: He makes two trips and carries $1.5 million on each trip. 6 2. Risk Aversion Continued a) Show which strategy Luke should pick if U=W .5 ? He is risk adverse. Under strategy B, EU=.25 U(3)+.5 U(1.5)+.25 U(0)=1.045. Under strategy A, EU=0.87. He is better off with strategy B. b) Show which strategy Luke should pick if U=W 2 ? Now he is a risk preferrer. Under strategy A, EU=4.5. Under strategy B, EU=25 U(3)+.5 U(1.5)+.25 U(0)=3.375. He is better off with strategy A.
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4 7 2. Risk Aversion Continued c) Discuss how the characteristics of Luke's utility functions in part a and b affect his choice between strategies . The choice depends on risk aversion.
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This note was uploaded on 12/08/2008 for the course ECON 101 taught by Professor Buddin during the Spring '08 term at UCLA.

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rev101b - 1. Diversification A stock market speculator is...

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